Difference schemes with point symmetries and their numerical tests

Abstract

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.

26 pages 7 figures

References in corpus (8)

• Continuous Symmetries of Difference Equations
• Lie point symmetries of difference equations and lattices
• Continuous symmetries of Lagrangians and exact solutions of discrete equations
• Lie symmetries of multidimensional difference equations
• On a Schwarzian PDE associated with the KdV Hierarchy
• A heat transfer with a source: the complete set of invariant difference schemes
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Cited by in corpus (19)

• A Group Theoretical Identification of Integrable Cases of the Liénard Type Equation $\ddot{x}+f(x)\dot{x}+g(x)=0$ : Part I: Equations having Non-maximal Number of Lie point Symmetries
• Symmetry preserving discretization of SL(2,R) invariant equations
• Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations
• Lie-point symmetries of the discrete Liouville equation
• Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
• The Korteweg-de Vries equation and its symmetry-preserving discretization
• Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations
• The Weyl realizations of Lie algebras and left-right duality
• First integrals of ordinary difference equations beyond Lagrangian methods
• Invariant finite-difference schemes with conservation laws preservation for one-dimensional MHD equations
• Invariant compact finite difference schemes
• Multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
• Invariant Conservation Law-Preserving Discretizations of Linear and Nonlinear Wave Equations
• First integrals of ordinary difference equations which do not possess a variational formulation
• On classification of Lie algebra realizations
• Transformation Groups on Real Plane and their Differential Invariants
• Symmetry Preserving Numerical Schemes for Partial Differential Equations and their Numerical Tests
• Algebraic discretization of time-independent Hamiltonian systems using a Lie-group/algebra approach